On infinite transformations with maximal control of ergodic two-fold product powers

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure-preserving transformations. A class of transformations is constructed such that for any subset R aS, a"e a (c) (0, 1) there exists T in this class such that T (p) x T (q) is ergodic if and only if a R. This contrasts with the finite measure-preserving case where T (p) x T (q) is ergodic for all nonzero p and q if and only if T x T is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer k, a family of rank-one infinite measure-preserving transformations is constructed which have ergodic index k, but infinite conservative index.